Q. 88 - Calculus

Question

On a long road trip, you are driving along a straight portion of Route 188. Suppose that t hours after entering Nevada your distance from the Donut Hole is $s\left(t\right) = -10t 2 - 40t + 120$ miles.

(a) How long will it take you to reach the Donut Hole after entering Nevada?

(b) Find your velocity v(t) as you drive toward the Donut Hole.

(c) Are you accelerating or decelerating as you approach the Donut Hole? At what rate?

Step-by-Step Solution

Answer

Ans:

Part (a). The time taken to reach the Donut Hole after entering Nevada is 2 hours.

Part (b). velocity v(t) : $s\text{'}\left(t\right)=-20t-40$

Part (c). acceleration a(t): $-20$ rate its decelerating while approaching the Donut Hole.

Step 1. Given information:

given, t hours after entering Nevada your distance from the Donut Hole is $s\left(t\right) =-10t^2-40t+120$

Step 2. Solving Part (a):

If it takes t hours after entering Nevada your distance from the Donut Hole is 

$s\left(t\right) =-10t^2-40t+120$

Time taken to reach the Donut Hole after entering Nevada will be,

as we are going back so, $s\left(t\right)=0$

$$\begin{gathered} \therefore -10t^2-40t+120=0 \\ dividing with 10 on both side, \\ \frac{-10t^2-40t+120}{10}=\frac{0}{10} \\ -t^2-4t+12=0 \\ -t^2-6t+2t+12=0 \\ -t(t+6)+2(t+6)=0 \\ (t+6)(-t+2)=0 \\ so, x=-6,2 \end{gathered}$$

Time is always taken as positive so the time taken to reach the Donut Hole after entering Nevada is 2 hours.

Step 3. Solving Part (b):

For finding the velocity we need to differentiate the given equation with dt once :

$$\begin{gathered} s\left(t\right) =-10t^2-40t+120 \\ \frac{ds\left(t\right)}{dt}=\frac{d}{dt}(-10t^2-40t+120) \\ s\text{'}\left(t\right)=2(-10)t-40 \\ s\text{'}\left(t\right)=-20t-40 \\ s\text{'}\left(t\right)=-20(t+1) \end{gathered}$$

Step 4. Solving Part (b):

For finding the acceleration we need to differentiate the given equation with dt twice :

$$\begin{gathered} s\text{'}\left(t\right)=-20t-40 \\ \frac{ds\text{'}\left(t\right)}{dt}=\frac{d}{dt}(-20t-40) \\ s\text{'}\text{'}\left(t\right)=-20 \\ its decelerating. \end{gathered}$$